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Chapter 2: Problem 25

Find the point on the line \(y=3 x+1\) in the \(x y\) -plane that is closest to the point (2,4) .

Short Answer

Expert verified
The point on the line \(y = 3x + 1\) closest to the point (2,4) is \((x, y) = \left(\frac{11}{10}, \frac{43}{10}\right)\).

Step by step solution

01

Write the distance expression

To compute the distance between two points in the xy-plane, we use the distance formula: \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Let us denote the point on the line as \((x, y)\) and the given point as \((2, 4)\). As the point \((x, y)\) lies on the line \(y = 3x + 1\), we get the distance squared as follows: \[ D^2 = (x - 2)^2 + (3x + 1 - 4)^2 \]
02

Differentiate the distance expression and set it to zero to find the critical points

To find the point on the line where the distance to (2,4) is minimized, we can differentiate our expression for distance with respect to x and set it equal to 0. A bit of simplification of our distance squared expression gives: \[ D^2 = (x - 2)^2 + (3x - 3)^2 \] Now differentiate this expression with respect to x: \[ \frac{dD^2}{dx} = 2(x - 2) + 2(3x - 3)(3) \] Equating the derivative to zero to find the critical points: \[ 2(x - 2) + 18(x - 1) = 0 \]
03

Solve the equation for x

Now, we will solve the equation for x: \[ 2(x - 2) + 18(x - 1) = 0\\ 2x - 4 + 18x - 18 = 0\\ 20x = 22 \] Dividing by 20, we get: \[ x = \frac{11}{10} \]
04

Calculate the y-coordinate of the point on the line

Now that we have the x-coordinate, let's determine the y-coordinate of the point by substituting the value of x in the given line equation \(y = 3x + 1\): \[ y = 3\left(\frac{11}{10}\right) + 1 \] So, we get: \[ y = \frac{33}{10} + 1 = \frac{43}{10} \]
05

Present the answer

The point on the line \(y = 3x + 1\) closest to the point (2,4) is \((x, y) = \left(\frac{11}{10}, \frac{43}{10}\right)\).

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Most popular questions from this chapter

Suppose \(s(x)=\frac{x^{2}+2}{2 x-1}\) (a) Show that the point (1,3) is on the graph of \(s\). (b) Show that the slope of a line containing (1,3) and a point on the graph of \(s\) very close to (1,3) is approximately -4 [Hint: Use the result of Exercise \(25 .]\)

Explain why the composition of two rational functions is a rational function.

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r(x))^{2} t(x) $$

Suppose \(p\) and \(q\) are polynomials of degree 3 such that \(p(1)=q(1), p(2)=q(2), p(3)=q(3),\) and \(p(4)=q(4) .\) Explain why \(p=q\).

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (s \circ r)(x) $$
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